Direct differentiation method for sensitivity analysis based on transfer matrix method for multibody systems

Abstract

Summary On one hand, the new version of transfer matrix method for multibody systems (NV‐MSTMM), has been proposed by formulating transfer equations of elements in acceleration level instead of position level as in the original discrete time transfer matrix method of multibody systems to study multibody system dynamics. This new formulation avoids local linearization and allows using any integration algorithms. On the other hand, sensitivity analysis is an important way to improve the optimization efficiency of multibody system dynamics. In this paper, a totally novel direct differentiation method based on NV‐MSTMM for sensitivity analysis of multibody systems is developed. Based on direct differentiation method, sensitivity analysis matrix for each kind of element is established. By assembling transfer matrices and sensitivity analysis matrices based on differentiation law of multiplication, the sensitivity analysis equation of overall transfer equation is deduced. The computing procedure of the proposed method is also presented. All these improvements as well as three numerical examples show that the direct differentiation method based on NV‐MSTMM is suitable for optimizing the dynamic sensitivity in multi–rigid‐body systems.

Hamiltonian direct differentiation and adjoint approaches for multibody system sensitivity analysis

“…The derivatives of h must be provided by an engineer dealing with a particular task, whereas the state derivatives of displacement, velocity, and reaction forces can be computed automatically, based on the mathematical model of a multibody system expressed in Equations (12). The direct derivative of these equations with respect to the design vector b reads as:…”

“…The intermediate step while solving such problem via optimization methods is a gradient computation. First, let us augment the functional (10) with the EOM (12) to get:…”

“…The quantities = (t), = (t), = (t), and = (t) are arbitrary Lagrange multipliers that enforce the system equations (12). Let us emphasize that the gradient of both expressions (10) and 22is the same, that is:…”

“…Equations (29)-(30) constitute an adjoint system to the equations of motion (12). By finding a unique solution of the adjoint system, one can reformulate the variation of performance measure (23) to be explicitly dependent on a variation of design variables.…”

“…11 Moreover, it has recently been applied for sensitivity analysis of multibody systems described by the transfer matrix method. 12 The authors have also utilized the method in dynamic balancing of a four-bar linkage. 13 On the other hand, the adjoint variable method exploits an approach imported from calculus of variations and optimal control of dynamical systems.…”